Final answer:
To find the relative extrema of the function f(x), we need to find the values of x where the derivative f'(x) equals zero or does not exist. By analyzing the equation (3 - 2x - x²)sin(2x - 3) = 0 and the continuity and differentiability of the function, we can determine the number of relative extrema on the given interval.
Step-by-step explanation:
To find the relative extrema of the function f(x), we need to find the values of x where the derivative f'(x) equals zero or does not exist. In this case, the derivative f'(x) is given by:
f'(x) = (3 - 2x - x²)sin(2x - 3)
To find the values of x where f'(x) equals zero, we set it equal to zero and solve for x:
(3 - 2x - x²)sin(2x - 3) = 0
Next, we need to check if the derivative exists at any values of x between -4 and 2 by looking for points where the function is not differentiable. We can do this by checking if the function is continuous and its derivative is continuous within the given interval.
Once we have found the values of x where the derivative equals zero or is undefined, we can determine the number of relative extrema of the original function f(x) on the interval -4 < x < 2.
By analyzing the equation (3 - 2x - x²)sin(2x - 3) = 0, we can find the solutions for x. By analyzing the continuity and differentiability of the function, we can determine the number of relative extrema on the given interval.